• Organizing data
• Developing theoretical principles
• Exploring hypotheticals
• Explaining observed patterns
• Guiding policy
• Making forecasts

• We have questions about the causes of things…
• …but Nature does not answer our questions directly.
• We must propose hypothetical answers to our questions.

• We have questions about the causes of things…
• …but Nature does not answer our questions directly.
• We must propose hypothetical answers to our questions.
• Models are nothing other than hypothetical answers to questions we ask.
• Models are necessarily partial answers, at best.

• It is relatively easy to ask difficult questions.
• It is harder, but still relatively easy, to propose creative hypothetical answers.

• It is relatively easy to ask difficult questions.
• It is harder, but still relatively easy, to propose creative hypothetical answers.
• It is often much more difficult to hear what the data have to say.

• It is relatively easy to ask difficult questions.
• It is harder, but still relatively easy, to propose creative hypothetical answers.
• It is often much more difficult to hear what the data have to say.
• Before the data will give a response, they must be asked in the proper way.
• Their responses are always equivocal, but some answers are better than others.
• One difficulty: how do we quantify this?
• Another difficulty: the answer we get depends on the question we ask.
  How do we avoid misunderstanding the reply?

In seeking to understand biological phenomena, models are tools for looking at the data.

They focus attention on the discrepancies between our hypotheses and reality.

How to quantify these discrepancies?

• The natural and optimal quantification is in terms of surprise: \(-\log{p}\)
• A better answer is one in which the data are less surprising.
• In this view, a model must be generative before Nature will respond.
• Thus, properly speaking, a model is a probability distribution.
• We must model the noise and the error.

• In seeking to infer causal mechanism in intact biological systems, dynamics are especially informative.
• Hypotheses about causal mechanisms can often readily be formalized as dynamical systems models.
• The problems of inference
  • How can we estimate unknown parameters?
  • How can we decide among competing models?
  • How can we extract maximum information from time series data?
• When do we know when we need a better model vs better data?

1. Markov models
2. Examples: cholera and pertussis
3. Partially observed Markov processes
4. Plug and play inference methods
5. Plug and play methods in practice
6. Conclusions and recommendations

What are the roles of seasonal and decadal climate drivers in the epidemiology of cholera?

What is the best vaccination strategy?

1. Roles of seasonal and decadal climate drivers
   • complex seasonality
   • multiennial cycles
2. Importance of bacteriophage in environment
3. Relative importance of human-human vs environmental transmission
4. Durations of vaccine- and infection-induced immunity

A large family of models:

• How well do each of the models explain the data?
• What insights can we gain into the system’s dynamic self-regulation?

These questions involve inescapable technical complications:

• nonlinearity
• stochasticity
• nonstationarity
• time-varying parameters

1. Roles of seasonal and decadal climate drivers
   • complex seasonality
   • multiennial cycles
2. Importance of bacteriophage in environment
3. Relative importance of human-human vs environmental transmission
4. Durations of vaccine- and infection-induced immunity

(Lavine, King, and Bjørnstad 2011)

(Domenech de Cellès, Magpantay, King, and Rohani 2016)

Why is pertussis resurgent?

Hypothetical answers include:

• Changes in vaccine efficacy
• Vaccine-driven pathogen evolution
• Increased circulation of congeneric pathogens
• Asymptomatic transmission
• Loss of natural immune boosting
• Increased surveillance sensitivity

• durability of vaccine-induced immunity
• relative efficacy of natural- and vaccine-derived immunity

\[\lambda(I_1,I_2,t) = \frac{\beta(t)\,(I_1+\theta\,I_2)+\bar{\beta}\,\iota}{N}\]

Post-vaccination infections are observed at a reduced rate, \(\eta\).

(Magpantay, Domenech de Cellès, Rohani, and King 2016)

• Can we estimate the nature and durability of vaccine-induced protection?
• Which models explain the data adequately?

(Magpantay et al. 2016)

(Domenech de Cellès, Magpantay, King, and Rohani 2018)

(Domenech de Cellès et al. 2018)

(Domenech de Cellès et al. 2018)

(Domenech de Cellès et al. 2018)

(Domenech de Cellès et al. 2018; Rohani, Zhong, and King 2010)

(Fumanelli, Ajelli, Manfredi, Vespignani, and Merler 2012; Mossong et al. 2008)

• Can we estimate the nature and durability of vaccine-induced protection?
• Which models explain the data adequately?
• Questions of parameter estimation and model selection

(Domenech de Cellès et al. 2018)

The key principle is that we must model the error.

• Measurement error
  • Error
  • Finite precision
• Process noise
  • Environmental trends and fluctuations
  • Unmodeled heterogeneities
  • Model misspecification
  • Unmodeled variables

(King, Nguyen, and Ionides 2016)

(King et al. 2016)

(King et al. 2016)

(King et al. 2016)

(King et al. 2016)

The full joint density is:

\[f_{X,Y}(x,y;\theta) = f_0(x_0;\theta)\,\prod_{n=1}^N\!f_{n}(x_n|x_{n-1};\theta)\,g_{n}(y_n|x_n;\theta).\]

The likelihood function is the marginal density for \(Y\), evaluated at the data:

\[ \begin{split} \mathcal{L}(\theta)&=f_{Y}(y^*_1,\dots,y^*_n;\theta)\\ &=\int f_{X,Y}(x_0,\dots,x_N,y^*_1,\dots,y^*N;\theta)\, dx_1\cdots dx_n. \end{split} \]

• computing the likelihood requires integrating over \(X_t\)
  for all \(t\)
• traditional solution—be very clever:
  • simplify the model to make it statistically tractable
  • devise and tune an inference algorithm (e.g., MCMC sampler) specifically to the model
• but: are we testing just our hypothesis or the statistical method also?
• it is perilous to invest too much time in one model
  • the Pygmalion problem
  • an antidote: the method of multiple working hypotheses

Definition: an algorithm has the plug-and-play property if it has no need to compute the latent process transition density.

Plug-and-play methods access the latent process model only via simulation.

This puts essentially no restrictions on the form of the models that can be entertained.

They are also called “simulation-based” methods.

(He, Ionides, and King 2010; King et al. 2016)

• Feature-based methods
  • Approximate Bayesian computation (ABC)
  • Nonlinear forecasting (NLF)
  • Attractor-reconstruction-based methods
  • Probe matching
  • Synthetic likelihood
• Full-information methods (likelihood based)
  • Sequential Monte Carlo (the particle filter)
  • Particle Markov chain Monte Carlo
  • Iterated filtering

Variants of all of these are available in the pomp software package (King et al. 2016; [https://kingaa.github.io/pomp/]) and elsewhere.

• The best model agrees with already published results
• The best model is the one that I can estimate before my thesis is due
• The best model is the one that best agrees with my preconceptions
• The best model is the one that makes my supervisor the happiest
• An identifiable model is better than a non-identifiable model
• The best model is the one with the narrowest confidence intervals

• How well does the model fit the data?
  • How does the likelihood compare to benchmark values?
  • Does the model fit well? (fitting best \(\ne\) fitting well )
• How does the model explain the data?
  • What aspects of the data are explained?
  • What aspects are ascribed to noise?
• Model adequacy
  • Are the data consistent with the model’s assumptions?
  • If the model were correct, how plausible are the data?

• Predictive power
  • Can the model make accurate out-of-sample predictions?
• What predictions to parameter estimates make about the results of independent studies?
• Fertility
  • Does the model make testable predictions?

\[\lambda(t) = \frac{\beta(t)\,(I_1+\theta\,I_2)+\bar{\beta}\,\iota}{N}\]

Post-vaccination infections are observed at a reduced rate, \(\eta\).

(Magpantay et al. 2016)

Interpretation: in vaccinated hosts, infections are mild to asymptomatic, yet equally infectious

Flat profiles indicate lack of information in the data relative to the question.

In effect, the data refuse to answer the question.

(Magpantay et al. 2016)

(King, Ionides, Pascual, and Bouma 2008)

(King et al. 2008)

(King et al. 2008)

Is it necessary that all infected individuals be equally infectious?
(King et al. 2008)

among symptomatic infections, case fatality: \(0.34\pm 0.2\)
duration of immunity \(1.5 \pm 0.7~\text{yr}\)
(King et al. 2008)

• Simulation-based methods make it possible to obtain answers to questions posed in the form of models that embody our precise questions.
• One need not carefully tailor the statistical algorithm to the model.
• It is perilous to invest too much time in one model: the Pygmalion problem.

• To obtain answers to our questions, we must pose them properly: as generative, stochastic models.
• Therefore, we must take care to model the noise.
• Simulation-based inference methods facilitate scientific investigation.
• Effective, maximally efficient inference methods are available.
• There is an intense need for further methodological development of such methods to improve computational efficiency and accommodate new data types.